## Petal-like modes in Porro prism resonators

Optics Express, Vol. 15, Issue 21, pp. 14065-14077 (2007)

http://dx.doi.org/10.1364/OE.15.014065

Acrobat PDF (409 KB)

### Abstract

A new approach to modeling the spatial intensity profile from Porro prism resonators is proposed based on rotating loss screens to mimic the apex losses of the prisms. A numerical model based on this approach is presented which correctly predicts the output transverse field distribution found experimentally from such resonators.

© 2007 Optical Society of America

## 1. Introduction

*et al*[1

1. G. Gould, S. Jacobs, P. Rabinowitz, and T. Shultz, “Crossed Roof Prism Interferometer,” Appl. Opt. **1**, 533–
534 (1962). [CrossRef]

2. I. Kuo and T. Ko, “Laser resonators of a mirror and corner cube reflector: analysis by the imaging method,”
Appl. Opt. **23**, 53–56 (1984). [CrossRef] [PubMed]

2. I. Kuo and T. Ko, “Laser resonators of a mirror and corner cube reflector: analysis by the imaging method,”
Appl. Opt. **23**, 53–56 (1984). [CrossRef] [PubMed]

6. I. Singh, A. Kumar, and O. P. Nijhawan, “Design of a high-power Nd:YAG Q-switched laser cavity,” Appl.
Opt. **34**, 3349–3351 (1995). [CrossRef] [PubMed]

2. I. Kuo and T. Ko, “Laser resonators of a mirror and corner cube reflector: analysis by the imaging method,”
Appl. Opt. **23**, 53–56 (1984). [CrossRef] [PubMed]

3. G. Zhou and L.W. Casperson, “Modes of a laser resonator with a retroreflecting roof mirror,” Appl. Opt. **20**,
3542–3546 (1981). [CrossRef] [PubMed]

8. Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov, and Y. M. Treivish, “Formation of fields in resonators with a
composite mirror consisting of inverting elements,” Sov. J. Quantum Electron **17**, 1040–1043 (1987). [CrossRef]

3. G. Zhou and L.W. Casperson, “Modes of a laser resonator with a retroreflecting roof mirror,” Appl. Opt. **20**,
3542–3546 (1981). [CrossRef] [PubMed]

8. Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov, and Y. M. Treivish, “Formation of fields in resonators with a
composite mirror consisting of inverting elements,” Sov. J. Quantum Electron **17**, 1040–1043 (1987). [CrossRef]

## 2. Porro resonator concept

**) looking towards the other (element**

*h***). On encountering a prism, the field inverts around the prism apex, and reverses its propagation direction, traveling back towards the opposite prism. The same inversion and reversal of propagation direction takes place again, and this sequence repeats on each pass. The prisms would essentially be treated as perfect mirrors but with a field inverting property.**

*a**t*(

*x*,

*y*):

*A*(

*x*,

*y*) describes the amplitude effects and φ(

*x*,

*y*) describes the phase effects of the prism. In the case of a Porro prism, the amplitude screen introduces losses not only at the edges of the element (transverse confinement), but also at the small but significant bevel along the apex where the prism surfaces meet. The phase screen allows the optical path length to vary as a function of the input position on the prism face, for example, to model errors in the prism angle or fabrication errors on the prism surfaces. With this approach, the diffractive effects of the prisms are taken into account, and the screens can be treated as intra-cavity elements that change the eigenmodes of a standard mirror-mirror resonator. In this paper we employ only the amplitude screen approach to model perfect prisms with high losses where the prism edges meet. The transfer function for the new prism model then includes only the amplitude effects,

*t*(

*x*,

*y*) =

*A*(

*x*,

*y*), and describes a high loss region along the apex of the prism, with 100% losses, and no losses elsewhere within the clear aperture of the element.

*Porro angle*. By way of example, we consider the case of α = 60°, as illustrated in Figs. 3(a)-3(e). In the analysis to follow the pertinent information is the location of the prism apexes, which we illustrate as solid lines 1 and 2 in Fig. 3(a), corresponding to elements

**and**

*h***in Fig. 1 respectively.**

*a**a priori*knowledge of how the mode will develop, and hence start with a ray located as shown in Fig. 3(a), traveling away from the viewer towards PP 2. We have chosen this location based on the assumption of high loss along the apexes, thus avoiding the apex zones. At PP 2 the ray is inverted about the prism apex, and travels back towards the viewer parallel to the optical axis as indicated in Fig. 3(b). At PP 1, the ray is inverted about the prism axis, and travels back towards PP 2 [Fig. 3(c)]. This process continues until the complete pattern is created [Fig. 3(e)], and the ray has returned to its starting position. This happens after three round trips. Clearly subsequent reflections simply duplicate the pattern. A second example is shown in Figs. 3(f)-3(j), where we illustrate the case of α = 30°. The same propagation rules apply so that eventually, after six complete round trips the pattern starts repeating itself. Clearly this approach correctly predicts the observed petal pattern formation often observed from such lasers, but this is based on

*a priori*knowledge and not physical reasoning. Also, this approach is only useful for limited Porro angles.

_{1}= (

*x*

_{1},

*y*

_{1}) with angular displacement given by ϕ

_{1}. The region of high loss is then simply a line passing through the origin with slope

*y*

_{1}/

*x*

_{1}. Without any loss of generality we will assume the resonator is viewed such that the first PP has an edge parallel to the horizontal axis, with the second PP rotated at the Porro angle, as illustrated in Fig. 1. It is easy to show that after

*n*reflections this vector has rotated through an angle θ

_{1}(

*n*) given by:

*n*reflections can be found from:

_{1}(0) = 0, so that if we imagine the apexes rotating about the unit circle, then the vector

*v*

_{1}(

*n*) may be expressed as:

*y*

_{1}= 0 and>

*y*

_{2}= (tanα)

*x*

_{2}respectively.

_{1}(

*n*) = ν

_{1}(0), which leads from Eq. (4) to the relation:

*i*. By selecting only the positive solutions for α one can derive a simple expression for the initial angles α that will lead to a finitely sub–divided field (or repeating pattern from the geometric viewpoint):

*i*and

*m*. The same result can be derived by starting from vector ν

_{2}. The implication is that only at these specific angles α will the field be finitely sub–divided, thus leading to some regions with low loss for lasing. In addition, since the position of these sub–divisions remains stable (i.e., they repeat on themselves) after a certain number of round trips, the modal pattern that oscillates inside such a resonator will give rise to a petal pattern

*only*at those angles given by Eq. (11). At other Porro angles the high loss apexes will continuously rotate to new positions, thus resulting in high losses across the entire field. We can now go on to calculate how many petals will be observed for a given Porro angle α. The number of petals will be equal to the number of sub–divisions of the field, but the field may not be completely sub–divided in one complete rotation of the vector; it may take several complete rotations for this to happen. We note that the sub–divisions will not necessarily be equal to the Porro angle; when several rotations of around the circle are needed to complete the sub–divisions, then the area between the initial apexes will be sub–divided further. In general we can write the following expression relating the Porro angle to the total number of sub–divisions (petals) of the field:

*N*must return the angle of each sub– division. If the sub–division is completed in one rotation, then the sub–division angle will equal α, but if more complete rotations are needed, then this will result in α itself being sub– divided by integer amount,

*j*. Thus both the left and right hand sides of Eq. (12) represent the same quantity - the final angle of each sub–division. A simple rearrangement of this equation then yields:

*N*must be an even number. The positive integer

*j*now appears to take on the meaning of the number of complete cycles required to return the apexes back onto one another. At present we cannot offer a simple analytical method of determining

*j*, but can offer the following conditions: (i)

*j*is the lowest positive integer such that

*N*is even, and (ii)

*j*≤ i.

*i*= 1; then

*j*= 1 and the circle is divided into divisions of α. For higher

*j*values the lossless regions between the high loss sub–division lines become small. Thus although there is an infinite number of solutions for α that lead to finite sub–divisions of the field, if the number of divisions is too large, diffraction will blur the spot structure and no petal pattern will be observed.

## 3. Test resonator

_{r}

^{4+}:YAG passive Q-switch. A quarter wave plate together with a polarizing beamsplitter cube ensured variable output coupling from the laser by polarization control (by rotation of the waveplate or by rotation of the prisms).

### 3.1 Experimental set-up

### 3.2 Numerical modeling

10. A. E. Siegman and H. Y. Miller, “Unstable Optical Resonator Loss Calculations Using Prony Method,” Appl.
Opt. **9**, 2729–2736 (1970). [CrossRef] [PubMed]

## 4. Results and discussion

## 5. Conclusion

## Appendix

## Acknowledgments

## References and links

1. | G. Gould, S. Jacobs, P. Rabinowitz, and T. Shultz, “Crossed Roof Prism Interferometer,” Appl. Opt. |

2. | I. Kuo and T. Ko, “Laser resonators of a mirror and corner cube reflector: analysis by the imaging method,”
Appl. Opt. |

3. | G. Zhou and L.W. Casperson, “Modes of a laser resonator with a retroreflecting roof mirror,” Appl. Opt. |

4. | J. Lee and C. Leung, “Beam pointing direction changes in a misaligned Porro prism resonator,” Appl. Opt. |

5. | Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev, and V. E. Sherstobitov, “Investigation of the properties of
an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser,” Sov. J. Quantum Electron. |

6. | I. Singh, A. Kumar, and O. P. Nijhawan, “Design of a high-power Nd:YAG Q-switched laser cavity,” Appl.
Opt. |

7. | N. Hodgson and H. Weber, |

8. | Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov, and Y. M. Treivish, “Formation of fields in resonators with a
composite mirror consisting of inverting elements,” Sov. J. Quantum Electron |

9. | T. A. Anan’ev, “Unstable prism resonators,” |

10. | A. E. Siegman and H. Y. Miller, “Unstable Optical Resonator Loss Calculations Using Prony Method,” Appl.
Opt. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3410) Lasers and laser optics : Laser resonators

(140.4780) Lasers and laser optics : Optical resonators

(230.5480) Optical devices : Prisms

(260.0260) Physical optics : Physical optics

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 23, 2007

Revised Manuscript: October 5, 2007

Manuscript Accepted: October 8, 2007

Published: October 11, 2007

**Citation**

Igor A. Litvin, Liesl Burger, and Andrew Forbes, "Petal–like modes in Porro prism resonators," Opt. Express **15**, 14065-14077 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-14065

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### References

- G. Gould, S. Jacobs, P. Rabinowitz and T. Shultz, "Crossed Roof Prism Interferometer," Appl. Opt. 1, 533-534 (1962). [CrossRef]
- I. Kuo and T. Ko, "Laser resonators of a mirror and corner cube reflector: analysis by the imaging method," Appl. Opt. 23, 53-56 (1984). [CrossRef] [PubMed]
- G. Zhou and L.W. Casperson, "Modes of a laser resonator with a retroreflecting roof mirror," Appl. Opt. 20, 3542-3546 (1981). [CrossRef] [PubMed]
- J. Lee and C. Leung, "Beam pointing direction changes in a misaligned Porro prism resonator," Appl. Opt. 27, 2701-2707 (1988). [CrossRef] [PubMed]
- Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977). [CrossRef]
- I. Singh, A. Kumar and O. P. Nijhawan, "Design of a high-power Nd:YAG Q-switched laser cavity," Appl. Opt. 34, 3349-3351 (1995). [CrossRef] [PubMed]
- N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005), Chap. 17.
- Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987). [CrossRef]
- T. A. Anan’ev, "Unstable prism resonators," Sov. J. Quantum Electron 3, 58-59 (1973).
- A. E. Siegman, H. Y. Miller, "Unstable optical resonator loss calculations using Prony Method," Appl. Opt. 9, 2729-2736 (1970). [CrossRef] [PubMed]

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